Expression For The Excess Pressure Due To Water Hammer
When a valve with a volumetric flow rate Q is closed, an excess pressure δP is created upstream of the valve, whose value is given by the Joukowsky equation:
In this expression:
- overpressurization δP is expressed in Pa;
- Q is the volumetric flow in m3/s;
- Zh is the hydraulic impedance, expressed in kg/m4/s.
The hydraulic impedance Zh of the pipeline determines the magnitude of the water hammer pulse. It is itself defined by:
with:
- ρ the density of the liquid, expressed in kg/m3;
- A cross sectional area of the pipe, m2;
- Beff effective modulus of compressibility of the liquid in the pipe, expressed in Pa.
The latter follows from a series of hydraulic concepts:
- compressibility of the liquid, defined by its adiabatic compressibility modulus Bl, resulting from the equation of state of the liquid generally available from thermodynamic tables;
- the elasticity of the walls of the pipe, which defines a modulus of equivalent compressibility Beq. In the case of a pipe of circular cross section whose thickness e is small compared to the diameter D, the equivalent modulus of compressibility is given by the following formula: ; in which E is the Young's modulus (in Pa) of the material of the pipe;
- possibly compressibility Bg of gas dissolved in the liquid, defined by:
- γ being the ratio of specific heats of the gas
- α the rate of ventilation (the volume fraction of undissolved gas)
- and P the pressure (in Pa).
Thus, the effective compressibility modulus is:
As a result, we see that we can reduce the water hammer by:
- increasing the pipe diameter at constant flow, which reduces the inertia of the liquid column;
- choosing to use a material with a reduced Young's modulus;
- introducing a device that increases the flexibility of the entire hydraulic system, such as a hydraulic accumulator;
- where possible, increasing the percentage of undissolved air in the liquid.
Read more about this topic: Water Hammer
Famous quotes containing the words expression, excess, pressure, due, water and/or hammer:
“They are not callow like the young of most birds, but more perfectly developed and precocious even than chickens. The remarkably adult yet innocent expression of their open and serene eyes is very memorable. All intelligence seems reflected in them. They suggest not merely the purity of infancy, but a wisdom clarified by experience. Such an eye was not born when the bird was, but is coeval with the sky it reflects. The woods do not yield another such a gem.”
—Henry David Thoreau (1817–1862)
“For most men, and most circumstances, pleasure—tangible material prosperity in this world—is the safest test of virtue. Progress has ever been through the pleasures rather than through the extreme sharp virtues, and the most virtuous have leaned to excess rather than to asceticism.”
—Samuel Butler (1835–1902)
“We believe that civilization has been created under the pressure of the exigencies of life at the cost of satisfaction of the instincts.”
—Sigmund Freud (1856–1939)
“From his childhood onwards this boy will be surrounded by sycophants and flatterers.... In due course, following the precedent which has already been set, he will be sent on a tour of the world and probably rumours of a morganatic marriage alliance will follow, and the end of it will be the country will be called upon to pay the bill.”
—James Keir Hardie (1856–1915)
“People always ask us, “Are things better or worse today?” Well, some things are better and some things are worse.... But there are a lot of problems in the world today that no one dreamed of when we were young. For instance, this business about the environment. Why, clean water was just something you took for granted.”
—Sarah Delany (b. 1890)
“This Administration has declared unconditional war on poverty and I have come here this morning to ask all of you to enlist as volunteers. Members of all parties are welcome to our tent. Members of all races ought to be there. Members of all religions should come and help us now to strike the hammer of truth against the anvil of public opinion again and again until the ears of this Nation are open, until the hearts of this Nation are touched, and until the conscience of America is awakened.”
—Lyndon Baines Johnson (1908–1973)